DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-30

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The induced tangent bundle chart

Definition

Let $(U, x)$ be a smooth chart on an $n$ -manifold $M$ . Every vector $v \in T_{p} M$ with $p \in U$ has a unique expansion $v = \sum_{i} v^{i} \partial_{x^{i}} ∣_{p}$ by Coordinate derivations form a basis of the tangent space. The induced tangent-bundle chart is $\tilde{x} : π^{- 1} (U) \to x (U) \times R^{n}, v \mapsto (x (p), (v^{1}, \dots, v^{n})) .$

The induced tangent bundle chart

Definition

Depends on

Dependency tree · 2 levels

Sources